1. Field of the Invention
The present invention relates to a lens system for focusing an image of an original or a light-emitting element as an erect image of an equal or unmagnified size on a photosensitive drum or a light detector in an optical copying machine, a facsimile, or an optical printer.
2. Description of the Relevant Art
FIG. 1 of the accompanying drawings shows a lens array 2 comprising a plurality of cylindrical lenses 1 each having a refractive index varying radially in a cross-sectional plane normal to an optical axis z. Each of the cylindrical lenses 1 has flat opposite end surfaces, i.e., flat entrance and exit end surfaces. The cylindrical lenses 1 are arrayed with their optical axes lying parallel to each other. The lens array 2, which serves to focus an erect image of an equal or unmagnified size, is widely used in the art. Each lens 1 has a radius r.sub.0 and a length z.sub.0.
The refractive index, n(r), of cylindrical lens 1 is given as a function of the radial distance r from the optical axis in a cross-sectional plane normal to the optical axis, as follows: EQU n.sup.2 (r)=n.sub.0.sup.2 [1-(gr).sup.2 ]. . . (1)
where n.sub.0 is the refractive index on the optical axis and g is a parameter indicating the gradient of the varying refractive index.
An erected unmagnified image of an object can be formed out of the lens under the condition: .pi.&lt;g.multidot.z.sub.0 &lt;2.pi.. When the distance from the object to the lens surface (entrance end surface) is infinite, g.multidot.z.sub.0 =.pi., and when the distance from the object to the lens surface is zero, g.multidot.z.sub.0 =2.pi..
The individual lenses 1 of the lens array 2 which employs such a graded-index medium and can focus an erect unmagnified image have focusing characteristics such that, with the refractive index varies as indicated by the equation (1), the on-axis spherical aberration is not corrected sufficiently as shown in FIGS. 2a and 2b, and the off-axis field curvature is large as shown in FIGS. 3a and 3b. As a result, the lens array 2 cannot focus a parallel beam of light into a small spot, and fails to achieve high resolution.
FIGS. 2a and 2b are diagrams of transverse aberration when FOB=0 and FIGS. 3b and 3b are diagrams of transverse aberration when FOB=0.8. In FIGS. 2a through 3b, FOB represents the fractional object height, and FPR the fractional pupil radius. The relationship between FOB, FPR, .DELTA.X, and .DELTA.Y is shown in FIGS. 11a and 11b.
.DELTA.Y indicates a Y component of a distance by which a ray R from an object O deviates from a chief ray Rc in an image plane in FIGS. 2a and 3a, and .DELTA.X indicates an X component of the distance in FIGS. 2b and 3b. The fractional object height FOB is given by: EQU FOB=OBY/OBYmax
where OBY is an object height and OBYmax is a maximum object height. The normalized pupil radius FPR is given by: EQU FPR=.rho./.rho.max
where .rho. is the distance of the ray R from an entrance pupil center P.sub.1 in an entrance pupil, and pmax is the maximum distance of the ray R from the entrance pupil center P.sub.1 in the entrance pupil, i.e., the radius of the entrance pupil.